1. Field of the Invention
The invention relates to a method and apparatus for calculating and visually displaying polarization states of an optical beam as it propagates through polarizing elements (e.g., waveplates and polarizers).
2. Description of the Prior Art
Polarization is one of the fundamental properties of electromagnetic radiation. Numerous investigations over the past two hundred years have sought to understand and model the state of polarization (SOP) of optical beams. This has led to dozens of applications of polarized light such as the measurement of the refractive index of optical materials, saccharimetry, ellipsometry, fluorescence polarization, etc., to name only a few. In recent years, fiber optic communications have led to new problems involving polarized light. Bit rates at and above 10 Gb/s can manifest polarization-related signal degradation caused by the birefringence of the fiber optic transmission medium. To mitigate these effects, it is important to measure, model and display the SOP of the optical beam.
There are several standard methods for modeling the SOP of an optical beam. One of the most useful is a polarimetric method known as the Poincarxc3xa9 Sphere (PS) method. This method is particularly valuable because it provides a quantitative visualization of the behavior of polarized light propagating through an optical fiber or optical polarization devices.
The Poincarxc3xa9 Sphere was suggested by Henri Poincarxc3xa9, a French mathematician, in the late 19th century. A goal of the Poincarxc3xa9, Sphere was to serve both as a visualization (display) tool and a calculating tool to describe polarized light as a polarized beam propagated through polarizing elements. In large part, the Poincarxc3xa9 Sphere is based on an analogy with a terrestrial (or celestial) sphere. For example, one can readily determine a distance between two locations, e.g., London and New York, by using equations of spherical trigonometry or by directly measuring arc length along a great circle between the two locations using a terrestrial globe and a tape measure. Poincarxc3xa9 conceived that SOP transformations performed by optical devices could be similarly quantified in terms of distances on the Poincarxc3xa9 Sphere.
Poincarxc3xa9 was motivated by near-intractability of direct calculations of SOP transformations using mathematics of his day. Nevertheless, his hoped-for simplicity using the sphere did not occur. It was an excellent visualization tool but most practical calculations using the sphere were still extremely difficult to do. Poincarxc3xa9 does not appear to have taken into account that no single conventional spherical polar coordinate system could simplify all polarization calculations.
Computation problems for polarized light were first solved in the late 1940s with an introduction of algebraic methods of the Jones and Mueller/Stokes calculi. These parametric calculi, however, did not directly enable simple visualizations of polarized light interactions. Thus, they did not fulfill Poincarxc3xa9""s goal of a device that would allow both visualization and calculation to be made in the same space, and without resort to complex algebraic and trigonometric calculations. Modern digital computers have automated the Jones/Mueller/Stokes computations, but this still does not provide a simple geometric representation of how polarized light behaves as it propagates.
Remarkably, a consistent mathematical treatment of the Poincarxc3xa9 sphere did not appear until H. Jerrard""s analysis in 1954, which provided some important clues about the problems with Poincarxc3xa9""s formulation. Jerrard wrote down the first formal algorithms for using the Poincarxc3xa9 sphere as a computing device and constructed a physical model to verify the usability of these algorithms. He found it necessary to mount a globe in a gimbal, with protractor markings, which could then be rotated with precision around both a north-south and an east-west axis. During computation, a reference point fixed in space above the surface of the sphere tracked the state of polarization, while the sphere was rotated underneath. The computational accuracy thus depended on mechanical stability, lack of eccentricity and manual dexterity.
To our knowledge, Jerrard""s implementation never came into use as a computational aid, due to its difficulties in both manufacture and use. Our analysis of its mechanical and operational complexity led back to Poincarxc3xa9""s original polar coordinate system which is optimally oriented for carrying out calculations involving rotational elements (polarizing rotators such as quartz rotators) but is not oriented for modeling phase shifting elements (waveplates).
Because of this limitation in the art, we developed a new polarization sphere, which we call the Observable Polarization Sphere (OPS). Through using the OPS, the polarization state of any beam propagating through a polarizing media, e.g., waveplate and linear polarizer, can be readily calculated and displayed. Consequently, the OPS can be used to calculate and display polarization behavior of any polarized or partially polarized beam as it propagates through an optical polarization system consisting of a series of polarizing media, e.g., waveplates or rotated linear polarizers.
Specifically, the present invention provides a method to visualize and calculate through, e.g., visual interpolation, the polarization behavior of an optical beam as it propagates through an optical system (e.g., fiber, bulk, or integrated). Through the OPS, all polarization computations are reduced to sequences of simple angular displacements along latitude lines and a prime meridian in the OPS coordinate system. Advantageously, elaborate mechanical contrivances previously needed to calculate phase shifts using the Poincarxc3xa9 Sphere are eliminated. The OPS demands only ordinary map-reading skills from its users.
With regard to specific methods of the invention itself, calculating the behavior of an optical system begins with determining a location of an input SOP in the OPS coordinate system. The SOP transformations are modeled as sequences of phase shifting and linear polarization operations starting from the initial input SOP, according to the following rules: (a) phase shifts are calculated by measuring out angular displacements (xcfx86) along latitudinal circles (constant xcex1); and (b) attenuation by a rotated linear polarizer is represented by a discontinuous jump to the north pole of the OPS, which is then followed by an angular displacement (xcex8) along the prime meridian (xcex4=0, xcfx80).
By concatenating a sequence of such angular displacements, effects of any sequence of waveplates and polarizers upon a beam of polarized light may be calculated. Each angular displacement is indicated visually by an arc segment plotted on a visible rendering of the OPS. The point on the OPS, that results after all the displacements have been measured out, advantageously represents a final SOP for the beam emerging from the optical system.